Most (but not all) very large companies I'm searching now don't have competitive ads. From what I can tell, the divide is not between public and private companies.

There is a box on the right with 'recent posts' and a big follow button that appears to be from Google circles.

The box in the right is in the upper right, and it's similar to Knowledge Graph entries but not the same. In most cases, it's a Google+ box, and the "follow button" seems to be just a link to the company's Google+ page. It's given apparently just to companies with very large Google+ followings.

"Recent posts" are recent Google+ posts, and there are variants of this, apparently depending on the number of followers. Try a search for [google] to see an example. On this day, April Fools Day, the 'recent post' links have image thumbnails and link to Google April Fools posts.

But it also appears that Google is experimenting with various combinations for other types of results. One result that I'm seeing is a public company with a local presence, and Google is showing what appears to be a cross between a Knowledge Graph and a Google+ local page, with a map, but also with a large thumbnail of a homepage video, and then the Knowledge Graph related searches. Chances are that this particular company doesn't have the Google+ numbers to justify a Google+ link button, but there is a link to a user review on the G+ Local page.

The ad exclusivity might be some sort of enhanced AdWords program test, related to the announcement discussed in this thread, which will allow others to bid on, but not display, your trademark in additional countries....